On robust theorems due to Bolzano, Weierstrass, Cantor, and Jordan

TL;DR

This paper explores the reverse mathematics of classical theorems by Bolzano, Weierstrass, Jordan, and Cantor, revealing new robust equivalences in higher-order systems that extend beyond the traditional 'Big Five' framework.

Contribution

It introduces two new series of equivalences in higher-order reverse mathematics related to classical theorems, demonstrating a richer logical landscape than previously known.

Findings

New equivalences are robust and distinct from the Big Five systems.

Higher-order reverse mathematics contains at least two additional 'Big' systems.

The equivalences are strictly between the base theory and higher-order weak König's lemma.

Abstract

Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics where the aim is to identify the minimal axioms needed to prove a given theorem from ordinary, i.e. non-set theoretic, mathematics. This program has unveiled surprising regularities: the minimal axioms are very often equivalent to the theorem over the base theory, a weak system of 'computable mathematics', while most theorems are either provable in this base theory, or equivalent to one of only four logical systems. The latter plus the base theory are called the 'Big Five' and the associated equivalences are robust following Montalban, i.e. stable under small variations of the theorems at hand. Working in Kohlenbach's higher-order RM, we obtain two long series of equivalences based on theorems due to Bolzano, Weierstrass, Jordan, and Cantor; these equivalences are extremely robust and have no counterpart among the Big Five systems, as they are strictly between the base theory and the higher-order counterpart of weak Koenig's lemma. Thus, higher-order RM is much richer than its second-order cousin, boasting (at least) two extra 'Big' systems.